The article was scanned from the
August 1971 American Aircraft Modeler (page 29). It has been out of print for decades,
and is difficult to access unless you are fortunate enough to buy one off of e-Bay. Hopefully
the original author won't mind that I have reproduced it here, but if he does, I will
remove it.

This is a very cleverly written article by Mr. John G. Burdick, and believe it or
not, there were some very unflattering - even angry - letters to the editor in the following
months.

There is one slightly nuanced feature that depends on the reader being familiar with
the meaning of complex numbers (those with both a real part and an imaginary part). If
you look at the graph in Figure 2, you will see in the caption that the scales have been
multiplied by a factor of √-1. According to what most people have been taught, one cannot
take the square root of a negative number. That is because if you take the result of
a square root and multiply it by itself, you should end up with your original number;
e.g., √4 = 2, so 2 * 2 = 4. That's good.

Now, suppose you try to do √-4, what number can you multiply by itself and end up
with -4? 2 * 2 = 4, and -2 * -2 = 4 also. Hmmm... So, in order to devise a way to work
with such entities, mathematicians came up with the "imaginary operator," i (or j in
engineering). We will just use i in the example. By definition, i is set equal to the
square root of -1. Therefore, if you say √-1 = i, then correspondingly, i * i = -1. Going
back to the example, then, √-4 = √(-1 * 4) = √-1 * √4 = i * 2 (written as √-4 = i2).
Now working back, i2 * i2 = (i * i) * (2 * 2) = -1 * 4 = -4. QED

Why go to that trouble to explain imaginary numbers? Look at the Figure 2 graph again.
If the torque and power scales are multiplied by √-1, that means the numbers are all
"imaginary;" not real! As I said, lots of people didn't get that part of it, and wrote
to condemn Mr. Burdick for being ignorant. The irony is great!

The Zero Displacement Engine

Fig. 2 - Predicted performance curves of ZDE. Torque is essentially
constant throughout RPM range. Power does not peak as with non-conventional engines.
For convenience in representation, Torque and Power scales shown have been multiplied
by a factor of √-1.

The specific power output of miniature aircraft engines has shown a remarkable increase
since the days of the Browns and Baby Cyclones. Probably the invention of the replaceable
glow plug was the most important of several technological factors contributing to this
increase, but advances in metallurgy, fuel chemistry and engine design have also been
contributory.

The author of this paper recently had occasion to plot from published data the relationship
between engine displacement and power output. Interestingly enough, this turned out to
be a non-linear plot; that is, engine output was not a simple multiple of engine displacement.
There was, rather, a definite tendency for the smaller engines to have a higher specific
power output than the larger engines. Curve-fitting techniques, using the least-squares
criterion, were applied to this data, using a regression equation of the form:

Power=Ad + Bd + C (where d equals displacement). A startling result emerged - C was
not equal to zero! In non-mathematical terms, this simply means that it should be possible
to construct a miniature engine having zero displacement but greater than zero power
output. This conclusion, while certainly surprising, is not altogether so; only a few
years ago the present 01 and 02 engines would have been regarded as impossible.

Given that a ZDE is theoretically possible, the remainder of this paper will concern
itself with the possible configurations and applications of such an engine.

Design of the ZDE: Since the product of two numbers is zero if either is zero, zero
displacement might be obtained through the choice either of zero stroke or zero bore.
Practical considerations, primarily the production problems inherent in manufacturing
a piston of zero diameter, suggest the choice of zero stroke rather than zero bore for
the ZDE. While this would definitely be counter to the current practice of square or
over-square design (that is, the bore-stroke ratio would be considerably smaller than
usual). it is felt that the advantages to be gained would outweigh this disadvantage.

Specifically, some of the advantages would be as follows. (1) Existing small engine
designs could be easily modified into the ZDE by proper crankshaft design: that is, with
the crankpin coaxial 'with the shaft proper. Provision for fuel induction would be similar
to current practice, except that the crankpin itself would be hollow. (2) Engine life
should be relatively great since even at high rpm's linear piston speed would be low.
(3) Very little dynamic crankshaft counterbalancing would be required. Note that factors
(2) and (3) taken together suggest that a very high operating rpm would be feasible,
perhaps in the 55,000-65,000 range.

Application of the ZDE: A reasonable estimate of the power output of the ZDE may be
obtained, if proper units are chosen, from the value of the constant C mentioned above.
This turns out to be about 0.025 hp, although the high operating rpm mentioned earlier
might lead to a somewhat greater power output. Since power outputs of this order would
seem ideal for indoor Re, discussion will be confined to this application.

Current RC power loading parameters suggest a weight of about six ounces as optimum
for an aircraft utilizing the ZDE. The recent development of extremely light radios suggest
that this is a practical figure and one which could be attained, or only slightly exceeded,
by careful construction technique. Certainly, the requirement for high-lift surfaces
which would lead to good low-speed performance, a primary requirement in indoor RC, would
also lead to the ability to tolerate a considerably higher power loading than usual.

A hypothetical design suitable for indoor ZDE-powered RC includes these design features:
a scale-like jet fuselage, necessary since the very high operating rpm of the ZDE, plus
its predictably low torque, would seem to dictate a shrouded propeller or ducted fan
design. The canopy provides room for the radio gear, since the cone necessary for ducted
fan operation would occupy most of the fuselage interior. The biplane design, utilizing
a high camber airfoil, would provide lift enough for good low-speed flight as well as
compensate for the possible high power loading. Finally, the counterbalanced control
surfaces were chosen to reduce actuator loads.

Conclusion: Several useful purposes would be served by the development of the ZDE,
primarily as a power unit for indoor RC and perhaps for indoor free-flight as well. It
is hoped that some manufacturer will. in the near future, consider making such an engine
available to model builders.